Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [
Citation: |
Figure 1. How the survival sets $ {\mathcal S}_k $ are nested. The pictures show preimages of $ \overline {\mathrm D}(0,2) $ at stages $ M_k $ (in red) and $ M_{k-1} $ (in blue) with $ m_k = 3 $. The dashed blue circle is $ {\mathrm C}(0,2) $ while the unit circle is shown in black. Observe how $ Q_{M_{k-1},M_k}^{-1}(\overline {\mathrm D}(0,2)) \subset \overline {\mathrm D}(0, 2) \setminus \overline {\mathrm D}(0, 1) $ as in Remark 1(c) is shown in red at Stage $ M_{k-1} $
Figure 2. Schematic for the proof of Theorem 1.6 in the case where $ \limsup |c_k| = +\infty $. Note how the round annulus $ {\mathrm A}(\sqrt{-c_{k}}, 1, \sqrt{|c_{k}|}) $ at stage $ M_{k-1} + m_k $ (in this case $ M_1+m_2 $) is pulled back conformally first by the preimage branches of $ Q_{M_{k-1},M_{k-1}+m_k} $ to form half the members of the collection $ \mathcal C $ at Stage $ M_1 $. Then the preimage branches of $ Q_{M_{k-1}} $ pull back the annuli in $ \mathcal C $ (one of which is visible in the zoomed box) to conformal annuli which separate the components of $ \mathcal{S}_k $ at stage $ 0 $
[1] |
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics.
![]() ![]() |
[2] |
Francisco Balibrea, On problems of topological dynamics in non-autonomous discrete systems, Appl. Math. Nonlinear Sci., 1 (2016), 391-404.
doi: 10.21042/AMNS.2016.2.00034.![]() ![]() ![]() |
[3] |
E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, FL, 2008.
![]() ![]() |
[4] |
L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4364-9.![]() ![]() ![]() |
[5] |
M. Comerford, A survey of results in random iteration, Proceedings Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), 435–476.
![]() ![]() |
[6] |
M. Comerford, Hyperbolic non-autonomous Julia sets, Ergodic Theory Dynamical Systems, 26 (2006), 353-377.
doi: 10.1017/S0143385705000441.![]() ![]() ![]() |
[7] |
A. Eremenko, Julia Sets are Uniformly Perfect, Preprint, Purdue University, 1992.
![]() |
[8] |
K. Falconer, Fractal Geometry, John Wiley & Sons, Ltd., Chichester, third edition, 2014. Mathematical foundations and applications.
![]() ![]() |
[9] |
J. E. Fornæss and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708.
doi: 10.1017/S0143385700006428.![]() ![]() ![]() |
[10] |
A. Hinkkanen, Julia sets of rational functions are uniformly perfect, Math. Proc. Cambridge Philos. Soc., 113 (1993), 543-559.
doi: 10.1017/S0305004100076192.![]() ![]() ![]() |
[11] |
S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., (4) (1996), 205-233.
![]() ![]() |
[12] |
R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc., 116 (1992), 251-257.
doi: 10.1090/S0002-9939-1992-1106180-2.![]() ![]() ![]() |
[13] |
C. T. McMullen, Complex Dynamics and Renormalization, Volume 135 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1994.
![]() ![]() |
[14] |
C. T. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., 20 (2010), 726-740.
doi: 10.1007/s00039-010-0078-3.![]() ![]() ![]() |
[15] |
L. Rempe-Gillen and M. Urbański, Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc., 368 (2016), 1979-2017.
doi: 10.1090/tran/6490.![]() ![]() ![]() |
[16] |
O. Sester, Hyperbolicité des polynȏmes fibrés, (French) [Hyperbolicity of fibered polynomials], Bull. Soc. Math. France, 127 (1999), 398-428.
![]() ![]() |
[17] |
R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575.
doi: 10.1090/S0002-9939-00-05313-2.![]() ![]() ![]() |
[18] |
R. Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, Ⅱ, Discrete Contin. Dyn. Syst., 32 (2012), 2583-2589.
doi: 10.3934/dcds.2012.32.2583.![]() ![]() ![]() |
[19] |
R. Stankewitz, H. Sumi and T. Sugawa, Hereditarily non uniformly perfect sets, Discrete Contin. Dyn. Syst S, 12 (2019), 2391-2402.
doi: 10.3934/dcdss.2019150.![]() ![]() |
[20] |
H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019.
doi: 10.1088/0951-7715/13/4/302.![]() ![]() ![]() |
[21] |
H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.
doi: 10.1017/S0143385701001286.![]() ![]() ![]() |
[22] |
H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922.
doi: 10.1017/S0143385705000532.![]() ![]() ![]() |
[23] |
H. Sumi, Dynamics of postcritically bounded polynomial semigroups Ⅲ: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles, Ergodic Theory Dynam. Systems, 30 (2010), 1869-1902.
doi: 10.1017/S0143385709000923.![]() ![]() ![]() |
[24] |
W. Zhiying, Moran sets and Moran classes, Chinese Sci. Bull., 46 (2001), 1849-1856.
doi: 10.1007/BF02901155.![]() ![]() ![]() |
How the survival sets
Schematic for the proof of Theorem 1.6 in the case where